Sturmian morphisms, the braid group B_4, Christoffel words and bases of F_2

We give a presentation by generators and relations of a certain monoid generating a subgroup of index two in the group Aut(F_2) of automorphisms of the rank two free group F_2 and show that it can be realized as a monoid in the group B_4 of braids on four strings. In the second part we use Christoffel words to construct an explicit basis of F_2 lifting any given basis of the free abelian group Z^2. We further give an algorithm allowing to decide whether two elements of F_2 form a basis or not. We also show that, under suitable conditions, a basis has a unique conjugate consisting of two palindromes.Comment: 25 pages, 4 figure

Estimating translational selection in Eukaryotic Genomes

Natural selection on codon usage is a pervasive force that acts on a large variety of prokaryotic and eukaryotic genomes. Despite this, obtaining reliable estimates of selection on codon usage has proved complicated, perhaps due to the fact that the selection coefficients involved are very small. In this work, a population genetics model is used to measure the strength of selected codon usage bias, S, in 10 eukaryotic genomes. It is shown that the strength of selection is closely linked to expression and that reliable estimates of selection coefficients can only be obtained for genes with very similar expression levels. We compare the strength of selected codon usage for orthologous genes across all 10 genomes classified according to expression categories. Fungi genomes present the largest S values (2.24–2.56), whereas multicellular invertebrate and plant genomes present more moderate values (0.61–1.91). The large mammalian genomes (human and mouse) show low S values (0.22–0.51) for the most highly expressed genes. This might not be evidence for selection in these organisms as the technique used here to estimate S does not properly account for nucleotide composition heterogeneity along such genomes. The relationship between estimated S values and empirical estimates of population size is presented here for the first time. It is shown, as theoretically expected, that population size has an important role in the operativity of translational selection